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Thread: Cube number

  1. #1
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    Cube number

    I'm wondering if there is any way to prove that of the numbers in the form 1.....1 (i.e. all digits are 1), 1 is the only cube number.
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    Quote Originally Posted by BenWong View Post
    I'm wondering if there is any way to prove that of the numbers in the form 1.....1 (i.e. all digits are 1), 1 is the only cube number.
    i was able to narrow it down a bit. i could prove that the only candidates for being a cube are the numbers of the form (10^{6m+1}-1)/9, \, (10^{6m+3}-1)/9, \, m \in \mathbb{Z}^+.
    i can post the proof of the above if you are familiar with modular arithmetic and fermat's little theorem.
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    That'd be great! I took abstract algebra a while ago, but I should be able to follow.
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    Quote Originally Posted by BenWong View Post
    That'd be great! I took abstract algebra a while ago, but I should be able to follow.
    first we use the fact that x^3 \equiv 0,\, 1, \, -1\,(\,mod\,7\,).
    proof: using fermat's li'l theorem x^6 \equiv 0, \,1 so either x \equiv 0 \, (\, mod \, 7 \, ) or x^6 \equiv 1 \,(\, mod \, 7 \,). the latter implies (x^3-1)(x^3+1) \equiv 0 \,(\,mod \, 7 \,)\Rightarrow x^3 \equiv 1,\,-1 \,(\,mod \, 7\,).

    so if (10^y-1)/9 has to be a cube then the inly admissible values of y would be the ones satisfying (10^y-1)/9 \equiv 0,\,1\,-1\,(\,mod\,7\,). there are only 7 trials needed to know which are those. the possible values of y are y=6m,\,6m+1,\,6m+3. that's it.
    in my last post i missed the numbers of the type (10^{6m}-1)/9.
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  5. #5
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    Quote Originally Posted by BenWong View Post
    I'm wondering if there is any way to prove that of the numbers in the form 1.....1 (i.e. all digits are 1), 1 is the only cube number.
    According to this book by Paulo Ribenboim, this result was proved by Andrzej Rotkiewicz in 1987. But I do not have a reference for the proof.
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  6. #6
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    So is the next step to show that a number 1....1 can never be written in those three forms. How do you show that?

    Thanks for the help.
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  7. #7
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by BenWong View Post
    So is the next step to show that a number 1....1 can never be written in those three forms. How do you show that?

    Thanks for the help.
    well i could not go further as i had already said "i could only narrow it down a bit".
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